Library / Geometry, coordinates, surveying and trigonometry / GEO-04
Geometry, coordinates, surveying and trigonometry
Area of a regular polygon from its perimeter
Area of a regular polygon from its full perimeter and its apothem.
- Ref
- GEO-04
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- A [cm²]
Splitting a regular polygon into n identical triangles with their apex at the centre, each has base s and height a, so the area is n × s × a/2. Since n s = P, the area is P a /2. The formula also holds for a circle, whose area is half the circumference times the radius.
Area of a regular polygon from its full perimeter and its apothem. Split into triangles with their apex at the centre: each has base s and height a.
Calculator
- Area of the circle with the same radius as the apothem, π a²28.27 cm²
- Share of that circle’s area1.273
- Perimeter of a circle of the same apothem, 2π a18.85 cm
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A perimeter of 24 cm and an apothem of 3 cm give 36 cm², which is 127 % of the circle that just fits inside.
- As the number of sides grows the polygon approaches that circle and the percentage approaches 100 % only if the perimeter equals 2π a.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (cm²) | Local step | Change in result |
|---|---|---|---|---|---|---|
| a Apothem | 0.5 to 20 cm | 6 | 240 | 234 | +10% | +10 % |
| P Perimeter | 4 to 120 cm | 6 | 180 | 174 | +10% | +10 % |
Sweep of Perimeter
| Perimeter (cm) | Area (cm²) |
|---|---|
| 4 | 6 |
| 18.5 | 27.75 |
| 33 | 49.5 |
| 47.5 | 71.25 |
| 62 | 93 |
| 76.5 | 114.8 |
| 91 | 136.5 |
| 105.5 | 158.3 |
| 120 | 180 |
Sweep one variable
Variables and how each one contributes
Area
What the equation gives you. Everything else feeds it.
Perimeter
The full distance round the polygon, in cm. A longer perimeter gives a larger area in direct proportion. Raising it by 10 % (to 26.4) moves the result by +10 %, from 36 to 39.6 cm².
Apothem
The apothem: the perpendicular distance from the centre to a side, in cm. A larger apothem gives a larger area in direct proportion. Raising it by 10 % (to 3.3) moves the result by +10 %, from 36 to 39.6 cm².
Worked example with the values above
- The perimeter is P = 24 cm and the apothem is a = 3 cm.
- Multiply: 24 × 3 = 72.
- A = 72 / 2 = 36 cm².
Assumptions and limits
- A flat (plane) shape with straight sides, measured in one consistent unit.
- For regular polygons all sides and all angles are equal.
Common mistakes
- Using a sloping side as the height when it is not perpendicular to the base.
- Mixing units, such as a base in mm and a height in cm.
- Using the diagonal of one shape in the formula for another.
Statistics and uncertainty
These are exact formulas, so the result is only as good as the lengths measured. Area scales with the square of a length, so a 1 % error in a length gives about a 2 % error in the area.
Where it is used
- Plate, panel and gasket areas from simple drawings.
- Checking the size of regular flanges, hex bars and tiles.
Maths you need
- Area: the flat space inside a shape, in square units
- Perpendicular: at a right angle to
- Tangent of an angle: opposite over adjacent